## Time domain analysis of optical amplification in Er^{3+} doped SiO_{2}-TiO_{2} planar waveguide

Optics Express, Vol. 13, Issue 12, pp. 4683-4692 (2005)

http://dx.doi.org/10.1364/OPEX.13.004683

Acrobat PDF (263 KB)

### Abstract

A time domain analysis of light amplification in an erbium doped silica-titania planar waveguide is reported. The investigation is performed by means of a home-made computer code which exploits the auxiliary differential equation scheme combined with the finite difference time domain technique to solve Maxwell’s equations and the rate equations. The simulation model takes into account the pump and input signal propagation, the secondary transitions pertaining to the ion-ion interactions and exploits the optical, spectroscopic and geometrical parameters measured on the fabricated waveguide.

© 2005 Optical Society of America

## 1. Introduction

^{3+}doped channel waveguide amplifiers (EDWA) have recently become very attractive in the field of the third window optical fiber communications. These devices show very interesting potential applications, overall for the possibility of integration with pump lasers and optical devices.

^{3+}concentration are necessary but in this way the concentration quenching effects become very considerable and detrimental to the optical gain, since they reduce the population in the first excited state [1

1. P.G. Kik and A. Polman, “Cooperative Upconversion as the Gain-Limiting Factor in Er Doped Miniature Al_{2}O_{3} Optical Waveguide Amplifiers,” J. Appl. Phys. **93**, 5008–5012 (2003). [CrossRef]

3. F. Di Pasquale and M. Federighi, “Modelling of Uniform and Pair-Induced Upconversion Mechanism in High-Concentration Erbium-Doped Silica Waveguides,” J. Lightwave Technol. **13**, 1858–1864 (1995). [CrossRef]

^{3+}ion excites a neighbouring unexcited Er

^{3+}ion, and of b) cooperative upconversion, in which an excited Er

^{3+}ion promotes a neighbouring excited Er

^{3+}ion into a higher lying state.

4. W.J. Miniscalco and R.S. Quimby, “General Procedure for Analysis of Er^{3+} Cross Section,” Opt. Lett. **16**, 258–260 (1991). [CrossRef] [PubMed]

5. A. D’Orazio, M. De Sario, L. Mescia, V. Petruzzelli, F. Prudenzano, A. Chiasera, M. Montagna, C. Tosello, and M. Ferrari, “Design of Er^{3+} Doped SiO_{2}-TiO_{2} Planar Waveguide Amplifier,” J. Non-Crystalline Solids **322**, 278–283 (2003). [CrossRef]

7. A. D’Orazio, V. De Palo, M. De Sario, V. Petruzzelli, and F. Prudenzano, “Finite Difference Time Domain Modeling of Light Amplification in Active Photonic Band Gap Structures,” Progress in Electromagnetics Research PIER **39**, 299–339 (2003). [CrossRef]

8. A.S. Nagra and R.A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Prop. **46**, 334–340 (1998). [CrossRef]

9. X. Jiang and C. M. Soukoulis, “Time Dependent Theory for Random Lasers,” Phys. Rev. Lett. **85**, 70–73 (2000). [CrossRef] [PubMed]

10. S. Chang and A. Taflove, “Finite-Difference Time Domain Model of Lasing Action in a Four-Level Two-Electron Atomic System,” Opt. Express **12**, 3827–3833 (2004). [CrossRef] [PubMed]

^{3+}doped SiO

_{2}-TiO

_{2}planar waveguide fabricated by rf sputtering [11

11. K. Jinguji, M. Horiguchi, S. Shibata, T. Kanamori, S. Mitachi, and T. Manabe, “Material Dispersion in Fluoride Glasses,” Electron. Lett. **18**, 164–165 (1982). [CrossRef]

12. C. Tosello, F. Rossi, S. Ronchin, R. Rolli, G.C. Righini, F. Pozzi, S. Pelli, M. Fossi, E. Moser, M. Montagna, M. Ferrari, C. Duverger, A. Chiappini, and C. De Bernardi, “Erbium-Activated Silica-Titania Planar Waveguides on Silica-on-Silicon Substrates Prepared by rf Sputtering,” J. Non-Crystalline Solids **284**, 243–248 (2001). [CrossRef]

_{2}-TiO

_{2}binary system as host material is due to the fact that it is possible to easily control the refractive index by changing the SiO

_{2}-TiO

_{2}molar ratio: this fact allows to optimize the amplifier performance.

## 2.Theory

_{2}-TiO

_{2}waveguide is performed by solving simultaneously a) the erbium rate equations which model the time evolution of the atomic energy level populations, b) the Maxwell equations and c) the auxiliary differential equation, i.e. the electron oscillator (EO) equation that takes into account the field effect on the medium during its propagation.

_{p}=980 nm, the typical transitions of the four-level laser systems occur. In particular, the ions at the ground state level I

_{15/2}(level 1) are excited to the level I

_{11/2}(level 3), the ground state absorption (GSA) cross-section having a peak at this wavelength. The optical gain, via the laser transition, I

_{13/2}→I

_{15/2}, occurs at the wavelength λ

_{a}=1532 nm, where the peak of stimulated emission (SE) takes place. The model, implemented in the ADE-FDTD computer code, includes the following other phenomena: (a) the up-conversion, Cup, due to ion pairs excited in the signal level I

_{13/2}(level 2), which results in the population of the I

_{9/2}level (level 4); (b) the up-conversion, C

_{3}, due to two neighbour ions excited in the pump level I

_{11/2}, which results in the population of the F

_{7/2}level; (c) the cross-relaxation, C

_{14}, between an ion excited in the level I

_{9/2}and an ion in the ground state I

_{15/2}which, by energy transfer, leave both in the level I

_{13/2}. The rate equations corresponding to the four-level system are the following:

_{i}are the population densities, the label i indicating the i-th level of the erbium ion, the concentration quenching coefficients C

_{up}and C

_{3}take into account the up conversion while C

_{14}is the cross relaxation coefficient, W

_{p}is the pumping rate,

**e**(t) is the electric field vector,

**p**(t) is the electric polarization vector, τ

_{ij}are the lifetimes between the i-th and j-th levels; the term

_{i}are linked by the conservation equation:

**d**, while

**h**is the magnetic field and µ

_{0}is the vacuum magnetic permeability.

**d**(t) in terms of the polarization

**p**(t) allows to link the Maxwell equations to the auxiliary differential equation derived by the classic electron oscillator (CEO) model, opportunely modified in order to be applied to a collection of resonant atoms:

_{0}the free space dielectric permeability. The polarization

**p**(t) has been expressed as the sum of two terms,

**p**

_{host}and

**p**

_{at}, corresponding to the polarization due to the host dielectric material and the resonant polarization produced by the laser atoms, respectively [13].

**p**

_{at}and the electric field for an isotropic medium are linked by the modified electron oscillator (MEO) equation:

_{1}and E

_{2}are the energy levels involved in the transition.

**p**

_{at}(t) to become much smaller than the value foreseen by the model of CEO, the total energy decay rate Δω

_{a}, which is the full width at half maximum (FWHM) linewidth of the atomic transition, is expressed by:

_{r}and γ

_{nr}are the energy radiative and non-radiative decay rates, respectively. T

_{2}is the mean time between de-phasing events.

_{osc}=γ

_{r}/(3γ

_{ceo}) is defined in the hypothesis that the atomic dipoles are fully aligned with the applied field. γ

_{ceo}is the energy decay rate of the CEO model,

*e*is the electron charge and

*m*is the electron mass.

_{12}=N

_{1}-N

_{2}is the electron population difference between the lower and upper energy levels involved in the transitions, N

_{1}and N

_{2}being the number of atoms per unit volume on two levels.

**P**

_{host}(ω)=ε

_{0}χ

_{host}

**E**(ω)

_{host}the host material electric susceptibility. Therefore the frequency domain constitutive relation can be so expressed:

_{host}is the host dielectric constant given by ε

_{host}=ε

_{0}[1+χ

_{host}].

_{at}(ω) for the resonant oscillator part in the laser medium can be obtained:

_{at}(ω) function is called complex Lorentzian lineshape.

_{at}(ω), the gain coefficient α

_{m}of the active medium is linked to the imaginary part ofχ

_{at}(ω). Furthermore the gain coefficient α

_{m}can be expressed in terms of the upward and downward cross sections.

_{12}(ω) and σ

_{21}(ω), it is possible to evaluate the imaginary part χ′ of the resonant electric susceptibility χ

_{at}(ω) as:

## 3. Numerical results

_{2}-TiO

_{2}core with transversal dimensions equal to 1.8 µm surrounded by a substrate of SiO

_{2}. The length of the device is opportunely fixed in order to obtain a gain as high as possible. The refractive indices of the core and substrate media have been evaluated via the Sellmeier equation [11

11. K. Jinguji, M. Horiguchi, S. Shibata, T. Kanamori, S. Mitachi, and T. Manabe, “Material Dispersion in Fluoride Glasses,” Electron. Lett. **18**, 164–165 (1982). [CrossRef]

5. A. D’Orazio, M. De Sario, L. Mescia, V. Petruzzelli, F. Prudenzano, A. Chiasera, M. Montagna, C. Tosello, and M. Ferrari, “Design of Er^{3+} Doped SiO_{2}-TiO_{2} Planar Waveguide Amplifier,” J. Non-Crystalline Solids **322**, 278–283 (2003). [CrossRef]

_{s}=1532 nm, are equal to n

_{core}=1.4650 and n

_{sub}=1.4452, respectively.

4. W.J. Miniscalco and R.S. Quimby, “General Procedure for Analysis of Er^{3+} Cross Section,” Opt. Lett. **16**, 258–260 (1991). [CrossRef] [PubMed]

_{eff}=1.445937 at the operating wavelength λ

_{s}=1532 nm. Moreover, the experimentally measured erbium emission and absorption cross sections [4

4. W.J. Miniscalco and R.S. Quimby, “General Procedure for Analysis of Er^{3+} Cross Section,” Opt. Lett. **16**, 258–260 (1991). [CrossRef] [PubMed]

_{at}(ω) (eq.17), have been used to evaluate the difference Δ

_{σ}=N

_{1}σ

_{12}(ω)-N

_{2}σ

_{21}(ω).

_{σ}has been fitted by means of five Lorentzian lineshape curves, the significant parameters of which are reported in Table 2. Figure 4 shows the Δ

_{σ}curve derived by the measured values and the perfectly reconstructed curve.

_{s}wavelength and the obtained results have been compared with those evaluated in the frequency domain [5

5. A. D’Orazio, M. De Sario, L. Mescia, V. Petruzzelli, F. Prudenzano, A. Chiasera, M. Montagna, C. Tosello, and M. Ferrari, “Design of Er^{3+} Doped SiO_{2}-TiO_{2} Planar Waveguide Amplifier,” J. Non-Crystalline Solids **322**, 278–283 (2003). [CrossRef]

_{t}is equal to 1.0498 while the frequency domain transmission coefficient T

_{f}is 1.0494. The same accuracy has been obtained when the concentration quenching effects have been neglected, i.e. the erbium energy system is reduced to a three-level system. As expected, the transmission coefficient improvement verifies when the concentration quenching phenomena are neglected. In fact in this case, for the same device length, we have obtained T

_{t}=1.0654 and T

_{f}=1.0653, respectively.

^{-12}s. After the pump signal application, the metastable energy level 2 becomes populated to the prejudice of the ground level 1. The energy levels 3 and 4 do not reach significant population densities because of their unstable nature. To consider the population inversion effectively verified, the quantity Δ

_{σ}must be positive and the population density N

_{2}must be equal to 95% of the starting population density N

_{1}. In the case under examination, only t=30 µs are sufficient to obtain the population inversion.

**e**(t)=sin(ω

_{s}(t-t

_{0}))exp(-((t-t

_{0})/ν)

^{2})

_{0}is the time at which the pulse is centered and ν is the pulse FWHM.

_{s}=1 µW, has been reported in Fig. 7 for three different values of the device length L=1 mm, L=2.5 mm and L=5 mm; it increases by increasing the pump signal power because of the increasing of the meta-stable energetic level population. For a pump signal power value of about P

_{p}=100 mW, the transmission coefficient becomes almost constant showing a saturation-like behaviour.

_{p}=300 mW and an input signal power P

_{s}=1 µW for the three device lengths. The transmission coefficient increases by increasing the erbium ion concentration; the effects of concentration quenching are not evident due to the considered short device lengths.

## 4. Conclusion

## Acknowledgments

## References and Links

1. | P.G. Kik and A. Polman, “Cooperative Upconversion as the Gain-Limiting Factor in Er Doped Miniature Al |

2. | M. Federighi, I. Massarek, and P.F. Trwoga, “Optical Amplification in Thin Optical Waveguides with High Er Concentration,” IEEE Photon. Technol. Lett. |

3. | F. Di Pasquale and M. Federighi, “Modelling of Uniform and Pair-Induced Upconversion Mechanism in High-Concentration Erbium-Doped Silica Waveguides,” J. Lightwave Technol. |

4. | W.J. Miniscalco and R.S. Quimby, “General Procedure for Analysis of Er |

5. | A. D’Orazio, M. De Sario, L. Mescia, V. Petruzzelli, F. Prudenzano, A. Chiasera, M. Montagna, C. Tosello, and M. Ferrari, “Design of Er |

6. | A. Taflove and S.C. Hagness, “ |

7. | A. D’Orazio, V. De Palo, M. De Sario, V. Petruzzelli, and F. Prudenzano, “Finite Difference Time Domain Modeling of Light Amplification in Active Photonic Band Gap Structures,” Progress in Electromagnetics Research PIER |

8. | A.S. Nagra and R.A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Prop. |

9. | X. Jiang and C. M. Soukoulis, “Time Dependent Theory for Random Lasers,” Phys. Rev. Lett. |

10. | S. Chang and A. Taflove, “Finite-Difference Time Domain Model of Lasing Action in a Four-Level Two-Electron Atomic System,” Opt. Express |

11. | K. Jinguji, M. Horiguchi, S. Shibata, T. Kanamori, S. Mitachi, and T. Manabe, “Material Dispersion in Fluoride Glasses,” Electron. Lett. |

12. | C. Tosello, F. Rossi, S. Ronchin, R. Rolli, G.C. Righini, F. Pozzi, S. Pelli, M. Fossi, E. Moser, M. Montagna, M. Ferrari, C. Duverger, A. Chiappini, and C. De Bernardi, “Erbium-Activated Silica-Titania Planar Waveguides on Silica-on-Silicon Substrates Prepared by rf Sputtering,” J. Non-Crystalline Solids |

13. | A.E. Siegman, “ |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(190.0190) Nonlinear optics : Nonlinear optics

(250.0250) Optoelectronics : Optoelectronics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 22, 2005

Revised Manuscript: May 24, 2005

Published: June 13, 2005

**Citation**

D. Biallo, A. D'Orazio, M. De Sario, V. Petruzzelli, and F. Prudenzano, "Time domain analysis of optical amplification in Er3+ doped SiO2-TiO2 planar waveguide," Opt. Express **13**, 4683-4692 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4683

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### References

- P.G. Kik, A. Polman, �??Cooperative Upconversion as the Gain-Limiting Factor in Er Doped Miniature Al2O3 Optical Waveguide Amplifiers,�?? J. Appl. Phys. 93, 5008-5012 (2003). [CrossRef]
- M.Federighi, I.Massarek, P.F.Trwoga, �??Optical Amplification in Thin Optical Waveguides with High Er Concentration,�?? IEEE Photon. Technol. Lett. 5, 227-229 (1993). [CrossRef]
- F. Di Pasquale, M. Federighi, �??Modelling of Uniform and Pair-Induced Upconversion Mechanism in High-Concentration Erbium-Doped Silica Waveguides,�?? J. Lightwave Technol. 13, 1858-1864 (1995). [CrossRef]
- W. J. Miniscalco, R. S.Quimby, �??General Procedure for Analysis of Er3+ Cross Section,�?? Opt. Lett. 16, 258- 260 (1991). [CrossRef] [PubMed]
- A. D�??Orazio, M. De Sario, L. Mescia, V. Petruzzelli, F. Prudenzano, A. Chiasera, M. Montagna, C. Tosello, M.Ferrari, �??Design of Er3+ Doped SiO2-TiO2 Planar Waveguide Amplifier,�?? J. Non-Crystalline Solids 322, 278-283 (2003). [CrossRef]
- A.Taflove, S.C.Hagness, �??Computational Electrodynamics: the Finite-Difference Time-Domain Method,�?? (Artech House Boston-London, 2000)
- A. D�??Orazio, V. De Palo, M. De Sario, V. Petruzzelli, F. Prudenzano, �??Finite Difference Time Domain Modeling of Light Amplification in Active Photonic Band Gap Structures,�?? Progress in Electromagnetics Research PIER 39, 299-339 (2003). [CrossRef]
- A. S. Nagra, R. A. York, �??FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,�?? IEEE Trans. Antennas Prop. 46, 334-340 (1998). [CrossRef]
- X. Jiang, C. M. Soukoulis, �??Time Dependent Theory for Random Lasers,�?? Phys. Rev. Lett. 85, 70-73 (2000). [CrossRef] [PubMed]
- S. Chang, A. Taflove, �??Finite-Difference Time Domain Model of Lasing Action in a Four-Level Two-Electron Atomic System,�?? Opt. Express 12, 3827-3833 (2004). [CrossRef] [PubMed]
- K. Jinguji, M. Horiguchi, S. Shibata, T. Kanamori, S. Mitachi, T. Manabe, �??Material Dispersion in Fluoride Glasses,�?? Electron. Lett. 18, 164-165 (1982). [CrossRef]
- C. Tosello, F. Rossi, S. Ronchin, R. Rolli, G. C. Righini, F. Pozzi, S. Pelli, M. Fossi, E. Moser, M. Montagna, M. Ferrari, C. Duverger, A. Chiappini, C. De Bernardi, �??Erbium-Activated Silica-Titania Planar Waveguides on Silica-on-Silicon Substrates Prepared by rf Sputtering,�?? J. Non-Crystalline Solids 284, 243-248 (2001). [CrossRef]
- A. E. Siegman, �??Lasers�?? (University Science Book 1986)

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